We review generalized zeta functions built over the Riemann zeros (in short:"superzeta" functions). They are symmetric functions of the zeros that displaya wealth of explicit properties, fully matching the much more elementaryHurwitz zeta function. As a concrete application, a superzeta function entersan integral representation for the Keiper--Li coefficients, whose large-orderbehavior thereby becomes computable by the method of steepest descents; thenthe dominant saddle-point entirely depends on the Riemann Hypothesis being trueor not, and the outcome is a sharp exponential-asymptotic criterion for theRiemann Hypothesis that only refers to the large-order Keiper--Li coefficients.As a new result, that criterion, then Li's criterion, are transposed to a novelsequence of Riemann-zeta expansion coefficients based at the point 1/2 (vs 1for Keiper--Li).
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